nLab G-∞-category

Idea

Let $\mathcal{S}_G$ be the (∞,1)-category of G-spaces, and let $\mathcal{O}_G$ be the orbit category. Elmendorf's theorem provides an equivalence $\mathcal{S}_G \simeq \mathrm{Fun}(\mathcal{O}_G^{\op},\mathcal{S})$. The starting point of Parametrized Higher Category Theory and Higher Algebra is to define $G$-$\infty$-categories so that they satisfy Elmendorf's theorem.

Definition

Definition

The ∞-category of small $G$-$\infty$-categories is

$$\mathrm{Cat}_{G,\infty} \coloneqq \mathrm{Fun}(\mathcal{O}_G^{\op}, \mathrm{Cat}_\infty).$$

More generally, often $\mathcal{T}$ will be an orbital ∞-category; in any case, we make the analogous definition.

Definition

If $\mathcal{T}$ is an (∞,1)-category, then the (∞,1)-category of small $\mathcal{T}$-$\infty$-categories is

$$\mathrm{Cat}_{\mathcal{T},\infty} \coloneqq \mathrm{Fun}(\mathcal{T}^{\op}, \mathrm{Cat}_\infty).$$

Examples

(Genuine) $G$-spaces

$G$-set induction furnishes an equivalence of categories $\mathbb{F}_H \xrightarrow\sim \mathbb{F}_{G,/[G/H]}$, which preserves and reflects transitivity; in particular, it restricts to an equivalence of orbit categories $\mathcal{O}_H \xrightarrow\sim \mathcal{O}_{G, [G/H]}$, with which we will conflate these two categories.

Using this, the universal fibration functor $\mathcal{O}_{-} = \mathcal{O}_{G, /-}:\mathcal{O} \rightarrow \Cat_{\infty}$ is a $G$-object in whose $H$-value is $\mathcal{O}_H$. By passing to presheaves of spaces fiberwise, we use this to define the $G$-$\infty$-category of $G$-spaces

$$\underline{\mathcal{S}}_G: \mathcal{O}_G^{\op} \xrightarrow{\;\;\;\; \mathcal{O}_{(-)} \;\;\;\; } \mathrm{Cat}_\infty^{\op} \xrightarrow{\;\;\;\; \mathrm{Psh}\;\;\;\; } \mathrm{Cat}_\infty.$$

Unwinding definitions, the following proposition is a form of Elmendorf's theorem.

Proposition

The $H$-value of the G-∞-category of $G$-spaces is the $\infty$-category $\mathcal{S}_H$ of H-spaces, and the induced functor $\Res_H^G:\mathcal{S}_G \rightarrow \mathcal{S}_H$ is restriction.

Thus $G$-functors out of $\underline{\mathcal{S}}_G$ are usually interpretable as collections of functors out of $(\mathcal{S}_H)$ which intertwine with restriction.

Related concepts

• G-space

• equivariant homotopy theory

• Parametrized Higher Category Theory and Higher Algebra, orbital ∞-category, equivariant symmetric monoidal category

References

• Clark Barwick, Emanuele Dotto, Saul Glasman, Denis Nardin, Jay Shah, Parametrized higher category theory and higher algebra: Exposé I – Elements of parametrized higher category theory, (arXiv:1608.03657)

• Jay Shah, Parametrized higher category theory, (arXiv:1809.05892)

• Jay Shah, Parametrized higher category theory II: Universal constructions, (arXiv:2109.11954)